simon-5502-05-slides

Topics to be covered

What you will learn
- Mathematical model of interactions
- Interactions in multi-factor analysis of variance
- Other examples of interactions
- Interactions in analysis of covariance
- Interactions in multiple linear regression

Mathematical model, 1

\(Y_{ijk}=\mu+\alpha_i+\beta_j+(\alpha \beta)_{ij}+\epsilon_{ijk}\)
- i=1,…,a j=1,…,b, k=1,…,n
If 1 is the reference category
- \(\alpha_1=0\)
- \(\beta_1=0\)
- \((\alpha \beta)_{1j}=0\)
- \((\alpha \beta)_{i1}=0\)

Interpretation

\(\mu\), estimated average outcome for both A, B at reference level
\(\alpha\) average change from reference level for A holding B fixed at reference level
\(\beta\) average change from reference level for B holding A fixed at reference level
\(\alpha \beta\) average deviation from parallelism

The interpretation of these parameters is a bit different when you have an interaction. The overall mean, mu, is the estimated average outcome when A and B are both at the reference level. This much has not changed.

The interpretation of the alpha values is different, but in a subtle way. It represents the average change when you switch A from the reference level to one of the other reference levels, holding B constant but constant at the reference level for B.

Similarly, beta represents the average change when you switch B from the reference level to one of the other reference levels, holding B constant at the reference level for B.

The interaction term, alpha-beta, is the degree to which a particular combination of non-reference levels for A and B differ from parallelism. Parallelism means a model with no interaction. You can think of the interaction term as the degree of super-additivity (if the term is positive) or sub-additivity (if the term is negative).

Mathematical model, 2

\(SS_A=\Sigma_i nb(\bar{Y}_{i..}-\bar{Y}_{...})^2\)
\(SS_B=\Sigma_j na(\bar{Y}_{.j.}-\bar{Y}_{...})^2\)
\(SS_{AB}=\Sigma_i \Sigma_j n(\bar{Y}_{ij.}-\bar{Y}_{i..}-\bar{Y}_{.j.}+ \bar{Y}_{...})^2\)
\(SS_E=\Sigma_i \Sigma_j \Sigma_k (Y_{ijk}-\bar{Y}_{ij.})^2\)
\(SS_T=\Sigma_i \Sigma_j \Sigma_k (Y_{ijk}-\bar{Y}_{...})^2\)

Test for an interaction

\(SS_{AB}\) has (a-1)(b-1) degrees of freedom
\(SS_E\) has ab(n-1) degrees of freedom
Accept \(H_0\) if \(F=\frac{MS_{AB}}{MS_E}\) is close to one
- In R, fit a model without an interaction
- Compare to a model with interaction
- Using the anova function

Break #1

What you have learned
- Mathematical model of interactions
What’s coming next
- Interactions in multi-factor analysis of variance

Hypothetical applications

Study with two drugs and two placebos
Four treatment options
- Placebo plus placebo
- Drug A plus placebo
- Placebo plus Drug B
- Drug A plus Drug B

Listing of hypothetical data

# A tibble: 8 × 3
  a     b         y
  <fct> <fct> <dbl>
1 P     P       0.9
2 P     P       1.1
3 A     P       2.9
4 A     P       3.1
5 P     B       3.9
6 P     B       4.1
7 A     B       9.9
8 A     B      10.1

Line plot of hypothetical data, 1

Create indicator variables

# A tibble: 8 × 6
  a     b       i_a   i_b  i_ab     y
  <fct> <fct> <dbl> <dbl> <dbl> <dbl>
1 P     P         0     0     0   0.9
2 P     P         0     0     0   1.1
3 A     P         1     0     0   2.9
4 A     P         1     0     0   3.1
5 P     B         0     1     0   3.9
6 P     B         0     1     0   4.1
7 A     B         1     1     1   9.9
8 A     B         1     1     1  10.1

Interaction model, 1

lm(y=a+b+a:b, data=hyp)
lm(y=a*b, data=hyp)
lm(y=i_a+i_b+i_ab, data=hyp_1)

Estimates from the hypothetical data

# A tibble: 4 × 2
  term        estimate
  <chr>          <dbl>
1 (Intercept)     1.00
2 i_a             2   
3 i_b             3   
4 i_ab            4

Interaction model, 2

(Intercept): estimated average outcome with two placebos
i_a: estimated average change due to drug A alone
i_b: estimated average change due to drug B alone
i_ab: estimated synergistic (or antagonistic) effect of both drugs

Interaction model, 3

P P mean = 1
A P mean = 1 + 2
P B mean = 1 + 3
A B mean = 1 + 2 + 3 + 4

The P P mean, the mean for both placebos, is the estimated intercept. The A P mean, the mean for drug A plus placebo, is the estimated intercept plus the estimate for the indicator for drug A. The P A mean, the mean for placebo plus drug B, is the estimated intercept plus the estimate for the indicator for drug B. The A B mean, the mean for drug A plus drug B, would be, if there was no interaction, the estimated intercept plus the estimate for the indicator for drug A plus the indicator for drug B. No interaction means that the combined effect of the two drugs is additive. With an interaction, you have to add in an extra term, the indicator for the interaction. This is a measure of the degree of super-additivity or sub-additivity. It is a measure of the degree of synergy or antagonism. In this case there is super-additivity or a lot of synergy.

Line plot of hypothetical data, 2

Line plot of hypothetical data, 3

Tracking a moving target

gender (M or F)
child (Yes or No)
trial4 (tracking time in seconds)

Means for each combination of gender and child

# A tibble: 4 × 3
  child gender mean_trial4
  <fct> <chr>        <dbl>
1 Yes   F             2.57
2 Yes   M             2.54
3 No    F             4.46
4 No    M             6.77

Interaction plot, 1

Interaction plot, 2

Regression estimates for the interaction model

# A tibble: 4 × 5
  term            estimate std.error statistic p.value  
  <chr>              <dbl>     <dbl>     <dbl> <glue>   
1 (Intercept)       2.57       0.382    6.73   p < 0.001
2 genderM          -0.0343     0.516   -0.0666 p = 0.947
3 childNo           1.89       0.654    2.89   p = 0.005
4 genderM:childNo   2.35       0.815    2.88   p = 0.005

intercept

intercept + genderMale

intercept + childNo

intercept + genderMale + childNo

intercept + genderMale + childNo + genderMale:childNo

Live demo, Interactions in multi-factor analysis of variance

Break #2

What you have learned
- Interactions in multi-factor analysis of variance
What’s coming next
- Other examples of interactions

Interaction of self-esteem and outcome

Plot showing an interaction between self-esteem and outcome

Interaction of training and autonomy

Interaction plot between training and autonomy

Interaction of group and gender

Interaction plot showing a gender difference but only in group 4

Live demo, Other examples of interactions

Break #3

What you have learned
- Other examples of interactions
What’s coming next
- Interactions in analysis of covariance

Interactions in analysis of covariance

When you suspect an interaction
- Graph your data
- Draw a separate trend line for each group
- Hope that the lines are mostly parallel
Compute models with and without the interaction
- Compare using the anova function
- Center your continuous covariate (very important!)

One of the assumptions of analysis of covariance is that there is no interaction between the covariate and the treatment/exposure variable. When the interaction is absent, you can totally remove the effect of the covariate.

An interaction changes things markedly. An interaction means that the effect of the covariate differs between the treatment group and the control group. You can no longer talk about removing the effect of the covariate by holding it constant. Instead, you have to recognize that the impact of the treatment varies, depending on the value of the covariate. A treatment that looks effective for some values of the covariate may be ineffective or even harmful for other values of the covariate.

If you suspect an interaction, first look at a graph. For the two levels of the treatment variable, draw a separate trend line associating the covariate with the outcome. If the lines are approximately parallel, that is a qualitative indication of no interaction.

A formal test of interaction requires you to fit a model with and without the interaction. Compare these two models using the anova function.

If you want to try to interpret the coefficients of the interaction model, you must (this is very important) center your covariate by subtracting the mean. Subtracting the mean places in the intercept in the heart of the data–no dangerous extrapolations needed.

Example: three exercise regimens

Two exercise regimens, jogging, swimmin
One control group: reading
- Subjects tracked hours per week (mean = 2)
Outcome variable, weight loss
- Negative values represent weight gain

Patients were put on two different exercise regimens, jogging or swimming, or in a control group that was asked to read for the same amount of time that the other two groups were supposed to spend exercising. You migt want to look at the effect of the three interventions after adjusting for the amount of time spent in each intervention.

The mean amount of time was 2 hours per week, so I computed a centerd version of hours by subtracing 2. negative values for this new variable meant that the subject spent less time on the intervention than the average subject. Positive values mean that the subject spent more time on the intervention than the average subject.

The intercept represents the estimated average weight loss for subjects with hours on the intervention equal to the overall average of 2. This is not an extrapolation. Most subejct spent somewhere between 1 and 4 hours on the intervention.

Visualizing the interaction

Testing the interaction

# A tibble: 2 × 7
  term                          df.residual    rss    df sumsq statistic p.value
  <chr>                               <dbl>  <dbl> <dbl> <dbl>     <dbl> <glue> 
1 loss ~ prog + hours_centered          896 43115.    NA   NA       NA   <NA>   
2 loss ~ prog + hours_centered…         894 37795.     2 5319.      62.9 p < 0.…

Coefficients and interpretation

# A tibble: 6 × 5
  term                        estimate std.error statistic p.value  
  <chr>                          <dbl>     <dbl>     <dbl> <glue>   
1 (Intercept)                    -3.70     0.376     -9.85 p < 0.001
2 progJogging                    11.8      0.531     22.3  p < 0.001
3 progSwimming                   29.6      0.531     55.7  p < 0.001
4 hours_centered                 -2.96     0.708     -4.18 p < 0.001
5 progJogging:hours_centered     10.4      1.07       9.71 p < 0.001
6 progSwimming:hours_centered     9.83     1.05       9.35 p < 0.001

Interpreting the intercept (-4)

Interpreting progJogging (12)

Interpreting progSwimming (30)

Interpreting hours_centered (-3)

Interpreting progJogging:hours_centered: (10)

Interpreting progSwimming:hours_centered: (10)

General conclusions

Jogging is better than reading, but only if you invest more than an hour per week.
Swimming is better than jogging or reading.
Extra time helps with the jogging and swimming interventions
Extra time hurts with the reading intervention

Break #4

What you have learned
- Interactions in analysis of covariance
What’s coming next
- Interactions in multiple linear regression

Interactions in multiple linear regression

Center your continuous predictors

Summary

What you have learned
- Mathematical model of interactions
- Interactions in multi-factor analysis of variance
- Other examples of interactions
- Interactions in analysis of covariance
- Interactions in multiple linear regression